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1.1 ÏòÁ¿¿Õ¼ä

1.(±±´ó2007).»Ø´ðÏÂÁÐÎÊÌ⣺

(1)ÊÇ·ñ´æÔÚn½×·½ÕóA,B,Âú×ãAB?BA?E (µ¥Î»¾ØÕó)£¿ÓÖ,ÊÇ·ñ´æÔÚnάÏßÐÔ¿Õ¼äÉϵÄÏßÐԱ任A,B,Âú×ãAB?BA?E (ºãµÈ±ä»»)£¿ÈôÊÇ,¸ø³öÖ¤Ã÷;Èô·ñ,¾Ù³öÀý×Ó.

(2) n½×ÐÐÁÐʽA¸÷ÐÐÔªËØÖ®ºÍΪ³£Êýc,ÔòAµÄ¸÷ÐÐÔªËØÖ®ºÍÊÇ·ñΪ³£Êý£¿ÈôÊÇ,ÊǶàÉÙ£¿ËµÃ÷ÀíÓÉ.

(3) m?n¾ØÕóÖÈΪr,È¡r¸öÏßÐÔÎ޹صÄÐÐÏòÁ¿,ÔÙÈ¡r¸öÏßÐÔÎ޹صÄÁÐÏòÁ¿,×é³ÉµÄr½××ÓʽÊÇ·ñÒ»¶¨Îª0£¿ÈôÊÇ,¸ø³öÖ¤Ã÷;·ñ,¾Ù³ö·´Àý.

(4) A,B¶¼ÊÇm?n¾ØÕó.ÏßÐÔ·½³Ì×éAX?0ÓëBX?0ͬ½â,ÔòAÓëBµÄÁÐÏòÁ¿ÊÇ·ñµÈ¼Û£¿ÐÐÏòÁ¿ÊÇ·ñµÈ¼Û£¿ÈôÊÇ,¸ø³öÖ¤Ã÷;·ñ,¾Ù³ö·´Àý.

32(5)°ÑʵÊýÓòR¿´³ÉÓÐÀíÊýÓòQÉϵÄÏßÐÔ¿Õ¼ä, b?pqr,ÕâÀïµÄp,q,rÊÇ»¥²»ÏàͬµÄËØÊý.ÅжÏ

3ÏòÁ¿×é1,nb,nb2,?,nbn?1ÊÇ·ñÏßÐÔÏà¹Ø£¿ËµÃ÷ÀíÓÉ.

2.(±±´ó2010).ÏòÁ¿×é?1,?2,?,?sÏßÐÔÎÞ¹Ø,ÇÒ¿ÉÒÔÓÉÏòÁ¿×é?1,?2,?,?lÏßÐÔ±í³ö.Ö¤Ã÷±Ø´æÔÚij¸öÏòÁ¿?j(j?1,2,?,l)ʹµÃÏòÁ¿×é?j,?1,?2,?,?sÏßÐÔÎÞ¹Ø.

3.(±±´ó2010).ÉèAÊÇn½×Õý¶¨¾ØÕó,ÏòÁ¿×é?1,?2,?,?sÂú×ã?iA?j?0(1?i?j?n).ÎÊ

'ÏòÁ¿×é?1,?2,?,?sµÄÖÈ¿ÉÄÜÊǶàÉÙ£¿Ö¤Ã÷ÄãµÄ½áÂÛ.

1.2 ÏßÐÔ·½³Ì×é

1.(±±´ó1997).ÉèA,BÊÇÊýÓòKÉϵÄn½×·½Õó, XÊÇδ֪Á¿x1,x2,?,xnËù³ÉµÄn?1¾ØÕó.ÒÑÖªÆë´ÎÏßÐÔ·½³Ì×éAX?0ºÍBX?0·Ö±ðÓÐl,m¸öÏßÐÔÎ޹ؽâÏòÁ¿,ÕâÀïl?0,m?0. (1)Ö¤Ã÷

(AB)X?0ÖÁÉÙÓÐmax(l,m)¸öÏßÐÔÎ޹ؽâÏòÁ¿. (2) Èç¹ûl?m?n, Ö¤Ã÷(A?B)X?0±ØÓзÇ

nÁã½â. (3)Èç¹ûAX?0ºÍBX?0ÎÞ¹«¹²·ÇÁã½âÏòÁ¿,ÇÒl?m?n, Ö¤Ã÷ KÖÐÈÎÒ»ÏòÁ¿?¿ÉΨһ

23

±í³É?????, ÕâÀï?,?·Ö±ðÊÇAX?0ºÍBX?0µÄ½âÏòÁ¿.

2.(±±´ó1998).ÌÖÂÛa,bÂú×ãʲôÌõ¼þʱ,ÊýÓòÉϵÄÏÂÊöÏßÐÔ·½³Ì×éÓÐΨһ½â,ÓÐÎÞÇî¶à¸ö½â,Î޽⣿µ±Óнâʱ,Çó³ö¸Ã·½³Ì×éµÄÈ«²¿½â.

?ax1?3x2?3x3?3,??x1?4x2?x3?1, ?2x?2x?bx?2.23?1l3.(±±´ó2001).Éè?ÊǸ´ÊýÓòCÉϵı¾Ô­´Îµ¥Î»¸ù(¼´, ?n?1,¶øµ±0?l?nʱ, ??1),

s,b¶¼ÊÇÕýÕûÊý,¶øÇÒs?n.Áî

?1?b?2b??2(b?1)?1?b?1A???????1?b?s?1?2(b?s?1)????. ???(n?1)(b?s?1)?????(n?1)b??(n?1)(b?1)??ÈÎÈ¡??CsÅжÏÏßÐÔ·½³Ì×éAX??ÓÐÎ޽⣿ÓжàÉٽ⣿д³öÀíÓÉ.

4.(±±´ó2006).(1)ÉèA,B·Ö±ðÊÇÊýÓòKÉÏs?n,s?m¾ØÕó.ÐðÊö¾ØÕó·½³ÌAX?BÓнâµÄ³ä·Ö±ØÒªÌõ¼þ,²¢¸øÓèÖ¤Ã÷.(2)ÉèAÊÇÊýÓòKÉÏs?nÁÐÂúÖȾØÕó,ÊÔÎÊ£º·½³ÌXA?EnÊÇ·ñÓн⣿ÈôÓнâ,д³öËüµÄ½â¼¯£ºÈôÎÞ½â,˵Ã÷ÀíÓÉ.(3) ÉèAÊÇÊýÓòKÉÏs?nÁÐÂúÖȾØÕó,ÊÔÎÊ£º¶ÔÓÚÊýÓòKÉÏ

s?m¾ØÕó,¾ØÕó·½³ÌAX?BÊÇ·ñÒ»¶¨Óн⣿µ±Óнâʱ,ËüÓжàÉÙ¸ö½â£¿Çó³öËüµÄ½â¼¯,ÒªÇó˵Ã÷ÀíÓÉ.

5.(±±´ó2008).»Ø´ðÏÂÁÐÎÊÌ⣺

(1) AÊÇs?n¾ØÕó.·ÇÆë´ÎÏßÐÔ·½³Ì×éAX??ÓнâÇÒrank(A)?r,ÔòAX??µÄ½âÏòÁ¿ÖÐÏßÐÔÎ޹صÄ×î¶àÓжàÉÙ¸ö£¿²¢ÕÒ³öÒ»×é¸öÊý×î¶àµÄÏßÐÔÎ޹ؽâÏòÁ¿. (2) AX??¶ÔÓÚËùÓеÄsά·ÇÁãÏòÁ¿?¶¼Óнâ,Çórank(A).

1??1??6.(±±´ó2010). ÉèA,BÊÇn½×¾ØÕó,ÇÒÂú×ãA??B?E??B?E?.Ö¤Ã÷: ¶ÔÈÎÒâµÄnά

110??110??ÁÐÏòÁ¿?,·½³Ì×éA(A?A)X?AT2TT?±ØÓзÇÁã½â.

n7.(ÖпÆÔº2007).Éè?1,?2,?,?k?RÊÇÆë´ÎÏßÐÔ·½³Ì×éAX?0µÄ»ù´¡½âϵ,

s,t?R, ?1?s?1?t?2,?,?k?1?s?k?1?t?k,?k?s?k?t?1.

ÊÔÎÊ£ºs,tÓ¦¸ÃÂú×ãʲô¹ØÏµ,ʹµÃ

?1,?2,?,?kÊÇ·½³Ì×éAX?0µÄ»ù´¡½âϵ,·´Ö®,µ±

24

?1,?2,?,?kÊÇ·½³Ì×éAX?0µÄ»ù´¡½âϵʱ,Õâ¸ö¹ØÏµ±ØÐë³ÉÁ¢.

8.(ÖпÆÔº2006).¿¼ÂÇÆë´ÎÏßÐÔ·½³Ì×éAX?0,ÆäÖÐA?(aij)(n?1)?n.ÉèMj(j?1,2,?,n)ÊÇÔÚϵÊý¾ØÕóAÖÐÏûÈ¥µÚjÁÐËùµÃµ½µÄn?1½××Óʽ.ÇóÖ¤: (1)M1,?M2,?,(?1)n?1MnÊÇ·½³Ì×éµÄÒ»¸ö½â;

(2)Èç¹ûAµÄÖÈΪn?1,ÄÇô·½³Ì×éµÄ½âÈ«ÊÇM1,?M2,?,(?1)n?1MnµÄ±¶Êý.

?????x1?x3?0,ÓÖ֪ijÏßÐÔÆë´Î·½³Ì×é(¢ò)µÄͨ½â9.(ÖпÆÔº2006).ÉèËÄÔªÆë´ÎÏßÐÔ·½³Ì×é(¢ñ)Ϊ?x?x?04?2Ϊk1(0,1,1,0)T?k2(?1,2,2,1)T. (1)ÇóÏßÐÔ·½³Ì×é(¢ñ)µÄ»ù´¡½âϵ;(2)ÎÊÏßÐÔ·½³Ì×é(¢ñ)ºÍ(¢ò)ÊÇ·ñÓзÇÁ㹫¹²½â£¿ÈôÓÐ,ÔòÇó³öËùÓеķÇÁ㹫¹²½â,ÈôûÓÐ,Ôò˵Ã÷ÀíÓÉ. 10.(ÖпÆÔº2004). ??xn?1?xn?4yn,ÒÑÖªx0?1,y0?0,Çóx100,y100.

y?2x?ynn?n?1?2x1?7x2?3x3?x4?5?x?3x?5x?2x?3?123411.(Öпƴó1997,2010).ÇóÏßÐÔ·½³Ì×é?µÄͨ½â.

?x1?5x2?9x3?8x4?1??5x1?18x2?4x3?5x4?1212.(Öпƴó1998).È¡ÄÄЩֵʱ,ÏÂÃæµÄ·½³Ì×éÓзÇÁã½â:

1?1??11???????11???x1??0???????1??x2??0????. ?????????????????n?1?????xn??0??1

1.3 ¾ØÕó´úÊý

1.(±±´ó2000).ÉèʵÊýÓòÉϵÄs?n¾ØÕóAµÄÔªËØÖ»ÓÐ0ºÍ1,²¢ÇÒAµÄÿһÐеÄÔªËØµÄºÍÊdz£Êý

r,AµÄÿÁ½¸öÐÐÏòÁ¿µÄÄÚ»ýΪ³£Êým,ÆäÖÐm?r. (1)Çó|AA'|;(2)Ö¤Ã÷s?n;(3)Ö¤Ã÷AA'µÄÌØ

Õ÷ֵȫΪÕýʵÊý.

2.(±±´ó2006).(1)ÉèA,B·Ö±ðÊÇÊýÓòKÉÏs?n,n?s¾ØÕó,Ö¤Ã÷£º

rank(A?ABA)?rank(A)?rank(En?BA).

(2) ÉèA,B·Ö±ðÊÇʵÊýÓòÉÏn½×¾ØÕó,Ö¤Ã÷£º¾ØÕóAÓë¾ØÕóBµÄÏàËÆ¹ØÏµ²»ËæÊýÓòÀ©´ó¶ø¸Ä±ä. 3.(±±´ó2007).¾ØÕóA,B¿É½»»»,Ö¤Ã÷rank(A?B)?rank(A)?rank(B)?rank(AB).

25

4.(±±´ó2008).(1)ÉèA,B·Ö±ðÊÇÊýÓòKÉÏs?n,n?m¾ØÕó,Ôò¶ÔÓÚËùÓÐm?l¾ØÕóC,ÊÇ·ñÓÐ

rank(ABC)?rank(BC)£¿¸ø³öÄãµÄÀíÓÉ.(2) AÊÇn½×¾ØÕó, AµÄÃ¿Ò»ÔªËØµÄ´úÊýÓà×Óʽ¶¼µÈ

ÓÚ´ËÔªËØ,Çórank(A).

5.(±±´ó2010).ÉèAÊÇ·ÇÁã¾ØÕó,Ö¤Ã÷A¿ÉÒÔд³Éij¸öÁÐÂúÖȾØÕóÓëij¸öÐÐÂúÖȾØÕóµÄ³Ë»ý. 6.(ÖпÆÔº2007).ÉèAÊÇn½×ʵÊý¾ØÕó,A?0,¶øÇÒAµÄÿ¸öÔªËØºÍËüµÄ´úÊýÓà×ÓʽÏàµÈ.Ö¤Ã÷AÊÇ¿ÉÄæ¾ØÕó.

?/n??1?7.(ÖпÆÔº2006).Èô?ΪһʵÊý,ÊÔ¼ÆËãlim?. ?n????/n1???a1???a???50100?100A8.(ÖпÆÔº2006).ÉèaΪʵÊý,A??ÇóµÄµÚÒ»ÐÐÔªËØÖ®ºÍ. ?R,??1???a???9.(ÖпÆÔº2004).ÉèA,BÊÇn½×ʵ·½Õó,¶øIÊÇn½×µ¥Î»Õó,Ö¤Ã÷:ÈôI?AB¿ÉÄæ,ÔòI?BAÒ²¿ÉÄæ.

n?10a???10.(ÖпÆÔº2003).ÒѸøÈçÏÂÈý½×¾ØÕó£ºA??01b?,(1)Çódet(A);(2)ÇóTr(A);(3)Ö¤Ã÷:

?cd1???rank(A)?2;(4)Ϊʹrank(A)?2,Çó³öa,b,cºÍdÓ¦Âú×ãµÄÌõ¼þ.

11.(ÖпÆÔº2010).(1)ÉèA,BÊÇn½×·½Õó, A¿ÉÄæ,BÃÝÁã,AB?BA.Ö¤Ã÷:A?B¿ÉÄæ; (2)ÊÔ¾ÙÀý˵Ã÷ÉÏÊöÎÊÌâÖÐA,B¿É½»»»µÄÌõ¼þ²»ÄÜÈ¥µô.

12.(Öпƴó1997).(1)Éèn½×¾ØÕóA????Ik?A21A12??,ÆäÖÐIkÊÇk½×µ¥Î»¾ØÕó,A22ÊÇn?k½×¾ØÕó.Ö¤Ã÷: A22??k?rank(A)?n,ÆäÖÐrank(A)ÊÇAµÄÖÈ.²¢Ö¤Ã÷rank(A)?kµÄ³äÒªÌõ¼þÊÇA22?A21A12.(2)ÉèAÊÇn½×¿ÉÄæ¾ØÕó, ?ºÍ?ÊÇnάÁÐÏòÁ¿,Ö¤Ã÷:n?1?rank(A???)?n,²¢ÇÒ

Trank(A???T)?n?1µÄ³äÒªÌõ¼þÊÇ: ?TA?1??1,ÕâÀï?T±íʾ?µÄתÖÃ.

?2?1?????12??13.(Öпƴó1997).Éè5½×3¶Ô½Ç¾ØÕóA??.(1)¼ÆËãAµÄÐÐÁÐʽdet(A); ????1????12???5?5

26

(2)ÇóAµÄÄæ¾ØÕóA;(3)ÇóAµÄJordan±ê×¼ÐÎ;(4)Çó¶Ô³Æ¾ØÕóAµÄÕý¡¢¸º¹ßÐÔÖ¸Êý;(5)½«½×Êý5¸ÄΪn,Çón½×·½ÕóAµÄÐÐÁÐʽºÍÄæ¾ØÕó. 14.(Öпƴó1998).¼ÆËã¾ØÕó£º

?1???cos7(1) ???sin??7??7??cos??7?sin??1997?1??0; (2) ?0??0?111??111?011??001???19.

15.(Öпƴó1999). n?2,n½×·½ÕóA?(aij)ÆäÖÐaij???0,i?j,?1Çódet(A)¼°A.

?1,i?j.16.(Öпƴó1999,2008).ÇóÖ¤:ÓëÈÎÒân½×·½Õó¿É½»»»µÄ·½ÕóÒ»¶¨ÊÇ´¿Á¿Õó.

¶þ¡¢ÐÐÁÐʽ

2.1 ¶¨Òå¡¢ÐÔÖʺͼÆËã·½·¨

1.(±±´ó2010).AÊǸ´¾ØÕó,BÊÇÃÝÁã¾ØÕó,ÇÒAB?BA.Ö¤Ã÷|A?2010B|?|A|. 2.(ÖпÆÔº2007).¼ÆËãn½×3¶Ô½ÇÐÐÁÐʽ

Dn?2cos?112cos??.

??112cos????????????n?n3.(ÖпÆÔº2006).ÒÑÖª?,?,?ΪʵÊý,ÇóA??µÄÐÐÁÐʽµÄÖµ. ?R????????????4.(ÖпÆÔº2005).¸ø¶¨Ò»µ¥µ÷µÝ¼õÐòÁÐb1?b2???bp?0,¶¨Òå????p!p?1????p?1?k?p?1?1min(bk?bk?1).

¼ÙÉ踴Êýak(k?1,2,?,p)Âú×ã|ak|??|ak?1|,k?1,2,?,p?1,ÇÒ|ap|?1.Ö¤Ã÷ÒÔÏÂÐÐÁÐʽ

a1b1b1a2D??1abpa1b2b2a2?2abp?a1p?a2p

???appbbbp1pbkbÆä¾ø¶ÔÖµÓÐÉÏϽçÈçÏ£º?|ak|?|D|?2?|ak|k.

pk?1k?1 27

5.(ÖпÆÔº2004).ÉèA?(aij)ÊÇ2004½×·½Õó,ÇÒaij?ij,1?i,j?2004.IÊÇ2004½×µ¥Î»Õó,¼ÆËã

f(x)?det(I?Ax),͉˕x?R.

6.(ÖпÆÔº2010).ÉèA,B·Ö±ðÊÇn?mºÍm?n¾ØÕó, IkÊÇk½×µ¥Î»¾ØÕó.(1)Ö¤Ã÷

det(In?AB)?det(Im?BA);

(2)¼ÆËãÐÐÁÐʽ

a1?x2?1?a1?x1?1?a2?x2?a?x1Dn?det?2????a?xan?x21?na1?xn???a2?xn??. ???1?an?xn???7.(ÖпÆÔº2011).Éèn½×·½ÕóAn??|i?j|?1?i,j?n,ÆäÐÐÁÐʽ¼ÇΪDn,ÊÔÖ¤Ã÷:

Dn?4Dn?1?4Dn?2?0.

²¢ÓÉ´ËÇó³öÐÐÁÐʽDn.

8.(Öпƴó2010).Ìî¿Õ£º(1)Éèn?2,A?(aij)n?n,aij?aj?biÔòdet(A)? .

a0??0??10?a1??(2)Éèn?1,¾ØÕóA????,ÔòAµÄÌØÕ÷¶àÏîʽÊÇ . ????0an?2????1an?1??

2.2 Ó¦ÓÃ

1.(ÖпÆÔº2003).¸øÁËn¸ö²»Í¬µÄÊýa1,a2,?,an,ÊÔÇóÒ»¸ö?n?1´ÎµÄ¶àÏîʽf(x),ʹ

f(ai)?bi,ÕâÀïbiÒ²ÊǸø¶¨µÄÖµ, i?1,2,?,n.

Èý¡¢ÏßÐÔ¿Õ¼äÓëÏßÐԱ任

3.1 ÏßÐÔ¿Õ¼äµÄ»ù±¾ÀíÂÛ

1.(±±´ó1996).ÉèÏßÐÔ¿Õ¼äVÖеÄÏòÁ¿×é

?1,?2,?3,?4ÏßÐÔÎÞ¹Ø. (1)ÊÔÎÊ£ºÏòÁ¿×é

?1??2,?2??3,?3??4,?4??1ÊÇ·ñÏßÐÔÎ޹أ¿ÒªÇó˵Ã÷ÀíÓÉ.(2)ÇóÏòÁ¿×é

28

?1??2,?2??3,?3??4,?4??1

Éú³ÉµÄÏßÐÔ×Ó¿Õ¼äWµÄÒ»¸ö»ùÒÔ¼°WµÄάÊý.

2.(±±´ó1998).ÉèVÊǶ¨ÒåÓòΪʵÊý¼¯RµÄËùÓÐʵֵº¯Êý×é³ÉµÄ¼¯ºÏ,¶ÔÓÚf,g?V,a?R·Ö±ðÓÃÏÂÁÐʽ×Ó¶¨Òåf?gÓëaf£º

(f?g)(x)?f(x)?g(x), (af)(x)?af(x), ?x?R.

ÔòV³ÉΪʵÊýÓòRÉϵÄÒ»¸öÏßÐÔ¿Õ¼ä. Éè

f0(x)?1,f1(x)?cosx,f2(x)?cos2x,f3(x)?cos3x.

(1)ÅжÏf0,f1,f2,f3ÊÇ·ñÏßÐÔÏà¹Ø,д³öÀíÓÉ;(2)ÓÃ?f,g?±íʾf,gÉú³ÉµÄÏßÐÔ×Ó¿Õ¼ä,ÅжÏ

?f0,f1???f2,f3?ÊÇ·ñΪֱºÍ,д³öÀíÓÉ.

3.(±±´ó1999).ÉèVÊÇʵÊýÓòRÉϵÄnάÏßÐÔ¿Õ¼ä, VÉϵÄËùÓи´Öµº¯Êý×é³ÉµÄ¼¯ºÏ,¶ÔÓÚº¯ÊýµÄ¼Ó·¨ÒÔ¼°¸´ÊýÓ뺯ÊýµÄÊýÁ¿³Ë·¨,Ðγɸ´ÊýÓòCÉϵÄÒ»¸öÏßÐÔ¿Õ¼ä,¼Ç×÷C. Ö¤Ã÷£ºÈç¹û

Vf1,f2,?,fn?1ÊÇCVÖÐn?1¸ö²»Í¬µÄº¯Êý,²¢ÇÒËüÃÇÂú×ã:

fj(???)?fj(?)?fj(?), ??,??V,

fj(k?)?kfj(?), ?k?R,??V,

Ôòf1,f2,?,fn?1ÊÇCÖÐÏßÐÔÏà¹ØµÄÏòÁ¿×é.

4.(ÖпÆÔº2006).ÈôÏòÁ¿?1,?2,?,?s(s?2)ÏßÐÔÎÞ¹Ø,ÌÖÂÛ?1??2,?2??3,?,?s?1??s,?s??1 µÄÏßÐÔÏà¹ØÐÔ.

5.(Öпƴó2010).Ìî¿Õ£º(1)Éè?1,?2,?3,?4ÊÇÏßÐÔ¿Õ¼äVÖÐ4¸öÏßÐÔÎ޹صÄÏòÁ¿,ÔòÏòÁ¿×é

V?1??2,?2??3,?3??4,?4??1µÄÖȵÈÓÚ . (2)ÔÚ3άʵÏòÁ¿¿Õ¼äR3ÖÐ,Éè

?1,?2,?3?ϵÄ×ø±ê?1?(?1,1,1)T,?2?(1,?1,0)T,?3?(1,0,?1)T,??(?4,3,4)T.Ôò?ÔÚ»ù?ÊÇ .

3.2 ÏßÐÔ¿Õ¼äµÄ×Ó¿Õ¼äºÍÉ̿ռä

1.(±±´ó1999).ÉèVÊÇÊýÓòKÉϵÄÒ»¸önάÏßÐÔ¿Õ¼ä, ?1,?2,?,?nÊÇVµÄÒ»¸ö»ù.ÓÃV1±íʾÓÉ

?1??2????nÉú³ÉµÄÏßÐÔ×Ó¿Õ¼ä,Áî

29

n?n?V2???ki?i:?ki?0,ki?K?.

i?1?i?1?(1)Ö¤Ã÷V2ÊÇVµÄ×Ó¿Õ¼ä;(2)Ö¤Ã÷V?V1?V2;(3)ÉèVÉϵÄÒ»¸öÏßÐԱ任AÔÚ»ù?1,?2,?,?nϵľØÕóAÊÇÖû»¾ØÕó(¼´: AµÄÿһÐÐÓëÿһÁж¼Ö»ÓÐÒ»¸öÔªËØÊÇ1,ÆäÓàÔªËØÈ«Îª0),Ö¤Ã÷V1ÓëV2¶¼ÊÇAµÄ²»±ä×Ó¿Õ¼ä.

2.(±±´ó2002).ÓÃR±íʾʵÊýÓò,¶¨ÒåRµ½RµÄÓ³ÉäÈçÏ£º

nf(X)?|x1|???|xr|?|xr?1|???|xr?s|, ?X?(x1,x2,?,xn)T?Rn,

nÆäÖÐr?s?0..Ö¤Ã÷£º(1)´æÔÚRµÄÒ»¸ön?rά×Ó¿Õ¼äW,ʹµÃf(X)?0,?X?W.(2)Èô

W1,W2ÊÇRnµÄÁ½¸ön?rά×Ó¿Õ¼ä,ÇÒÂú×ãf(X)?0,?X?W1?W2,ÔòÒ»¶¨ÓÐ

dim(W1?W2)?n?(r?s).

3.(±±´ó2002).ÉèVÊÇÊýÓòKÉϵÄnάÏßÐÔ¿Õ¼ä, V1,?,VsÊÇVµÄs¸öÕæ×Ó¿Õ¼ä,Ö¤Ã÷£º(1)´æÔÚ

??V,ʹµÃ??V1?V2???Vs.(2)´æÔÚVÖеÄÒ»×é»ù?1,?2,?,?n,ʹµÃ

{?1,?2,?,?n}??V1?V2???Vs???.

4.(±±´ó2005).ÉèÊýÓòKÉϵÄn¼¶¾ØÕóAµÄ(i,j)ԪΪai?bj £¨1£©.ÇóA; (2).µ±n?2ʱ£¬

a1?a2,b1?b2.ÇóÆë´ÎÏßÐÔ·½³Ì×éAX?0µÄ½â¿Õ¼äµÄάÊýºÍÒ»¸ö»ù.

5.(±±´ó2005).£¨1£©ÉèÊýÓòKÉÏn¼¶¾ØÕ󣬶ÔÈÎÒâÕýÕûÊým£¬ÇóC [CÊÇʲô£¿]

£¨2£©ÓÃMn(K)±íʾÊýÓòKÉÏËùÓÐn¼¶¾ØÕó×é³ÉµÄ¼¯ºÏ£¬Ëü¶ÔÓÚ¾ØÕóµÄ¼Ó·¨ºÍÊýÁ¿³Ë·¨³ÉΪKÉÏ

ma1µÄÏßÐԿռ䡣ÊýÓòKÉÏn¼¶¾ØÕóA?a2a3?anana1a2?an?1³ÆÎªÑ­»·¾ØÕó¡£ÓÃU±íʾKÉÏËùÓÐn?????a2a3a4?a1¼¶Ñ­»·¾ØÕó×é³ÉµÄ¼¯ºÏ¡£Ö¤Ã÷£ºUÊÇMn(K)µÄÒ»¸ö×ӿռ䣬²¢ÇóUµÄÒ»¸ö»ùºÍάÊý¡£ 6.(ÖпÆÔº2003).¸ø¶¨RÉÏÏßÐÔ¿Õ¼äVµÄ×Ó¿Õ¼äW1,W2,Ö¤Ã÷:

dim(W1?W2)?dim(W1)?dim(W2)?dim(V),

30

ÕâÀïdim±íʾ¿Õ¼äάÊý. 7.(2009Ê¡ÇøÈü).ÉèCn?nÊÇn?n¸´¾ØÕóÈ«ÌåÔÚͨ³£µÄÔËËãÏÂËù¹¹³ÉµÄ¸´ÊýÓòCÉϵÄÏßÐÔ¿Õ¼ä,

?0??1F??0???0?0001?0?????0?an??n?10?an?1?n?1?kn?n, Æä0?an?2?. (1)¼ÙÉèA??aij?n?n?C,ÈôAF?FA,Ö¤Ã÷:A??an?k,1Fk?0???1?a1??n?nÖÐ, F?E(n½×µ¥Î»¾ØÕó). (2)ÇóCµÄ×Ó¿Õ¼äC(F)?X?Cn?n:FX?XFµÄάÊý.

??

3.3 ÏßÐÔÓ³ÉäÓëÏßÐԱ任

1.(±±´ó1996).ÉèVÊÇÊýÓòKÉϵÄnάÏßÐÔ¿Õ¼ä,²¢ÇÒV?U?W,Èθø??V,Éè

???1??2,?1?U,?2?W,ÁîP(?)??1.Ö¤Ã÷:(1) PÊÇVÉϵÄÏßÐԱ任,²¢ÇÒP2?P;;(2)

PµÄºËkerP?W, PµÄÏñ(ÖµÓò) ImP?U;(3) VÖдæÔÚÒ»¸ö»ù,ʹµÃPÔÚÕâ¸ö»ùϵľØÕóÊÇ

?Ir??O?O??,ÆäÖÐIr±íʾr¼¶µ¥Î»¾ØÕó,ÇëÖ¸³öµÈÓÚʲô£¿ O??2.(±±´ó1999). ÉèVºÍUÊÇÊýÓòKÉϵÄnά, mάÏßÐÔ¿Õ¼ä, AÊÇVµ½UµÄÒ»¸öÏßÐÔÓ³Éä.(1)Ö¤Ã÷£ºdim(KerA)?dim(ImA)?dimV;(2)Ö¤Ã÷:Èç¹ûdimV?dimU,ÔòAÊǵ¥Éäµ±ÇÒ½öµ±ÊÇAÂúÉä. 3.(±±´ó2000).ÉèVºÍV'ÊÇÊýÓòKÉϵÄÓÐÏÞάÏßÐÔ¿Õ¼ä, AÊÇVµ½V'µÄÒ»¸öÏßÐÔÓ³Éä.Ö¤Ã÷,´æÔÚÖ±ºÍ·Ö½â: V?U?W,V'?M?N,ʹµÃKerA?U,²¢ÇÒW?M.

4.(±±´ó2001).ÉèAÊÇÊýÓòKÉÏnάÏßÐÔ¿Õ¼äÉϵÄÒ»¸öÏßÐԱ任.ÔÚK[x]ÖÐ,

f(x)?f1(x)f2(x),ÇÒf1(x)Óëf2(x)»¥ËØ,ÓÃKerA±íʾÏßÐԱ任AµÄºË.Ö¤Ã÷£º

Kerf(A)?Kerf1(A)?Kerf2(A).

5.(±±´ó2001).ÉèAÊÇÊýÓòKÉÏnάÏßÐÔ¿Õ¼äÉϵÄÒ»¸öÏßÐԱ任.ÊÇIºãµÈ±ä»».Ö¤Ã÷£ºA?AµÄ³ä·Ö±ØÒªÌõ¼þÊÇ rank(A)?rank(A?I)?n.

6.(±±´ó2002).ÉèÕýÕûÊýn?2,ÓÃMn(K)±íʾÊýÓòKÉÏÈ«Ìån?n¾ØÕó¹ØÓÚ¾ØÕó¼Ó·¨ºÍÊý³ËËù¹¹³ÉµÄKÉϵÄÏßÐÔ¿Õ¼ä.ÔÚMn(K)Öж¨Òå±ä»»?ÈçÏ£º

2 31

???aij?n?n??aij?n'' ÆäÖÐ a??(a)?M(K),?aij,?i?aijijn?nnn?n?ii??i?1??i?j;i?j.

(1)Ö¤Ã÷?ÊÇMn(K)ÉϵÄÏßÐԱ任.(2)Çó³öKer(?)µÄάÊýÓëÒ»×é»ù.(3)Çó³ö?µÄÈ«²¿ÌØÕ÷×Ó¿Õ¼ä.

7.(±±´ó2005).ÉèAÊÇÊýÓòRÉÏnάÏßÐÔ¿Õ¼äVÉϵÄÒ»¸öÏßÐԱ任£¬ÓÃI±íʾVÉϵĺãµÈ±ä»»£¬Ö¤Ã÷£º A3?I?rank(I?A)?rank(I?A?A2)?n

8.(±±´ó2006).ÉèVÊÇÊýÓòKÉϵÄnάÏßÐÔ¿Õ¼ä, A1,A2,?,AsÊÇVÉϵÄÏßÐԱ任,Áî

A?A1?A2???As.Ö¤Ã÷:

AΪÃݵȱ任ÇÒ rank(A)?rank(A1)?rank(A2)???rank(As)

µÄ³ä·Ö±ØÒªÌõ¼þÊǸ÷¾ùΪAiÃݵȱ任,ÇÒAiAj?0,i?j.

9.(±±´ó2007). nάÏßÐÔ¿Õ¼äVÉϵÄÒ»¸öÏßÐԱ任AµÄ×îС¶àÏîʽÓëÌØÕ÷¶àÏîʽÏàͬ.ÊÔÖ¤Ã÷´æÔÚ??V,ʹµÃ

??,A?,?,A??ΪVµÄÒ»¸ö»ù.

n?110.(±±´ó2008).ÉèVÊÇÊýÓòKÉϵÄnάÏßÐÔ¿Õ¼ä, A,BÊÇVÉϵÄÏßÐԱ任,ÇÒA,BµÄ×îС¶àÏîʽ»¥ËØ,ÇóÂú×ãAC?CBµÄËùÓÐÏßÐԱ任C.

n11.(ÖпÆÔº2006).ÉèfÊÇÓÐÏÞάÏòÁ¿¿Õ¼äVÉϵÄÏßÐԱ任,ÇÒfÊÇVÉϵĺãµÈ±ä»»,ÕâÀïnÊÇij

¸öÕýÕûÊý.ÉèW?{v?V:f(v)?v}.Ö¤Ã÷WÊÇVµÄÒ»¸ö×Ó¿Õ¼ä,²¢ÇÒÆäάÊýµÈÓÚÏßÐԱ任

1(f?f2???fn)µÄ¼£. n12.(ÖпÆÔº2004).ÉèVÊÇnάÏòÁ¿¿Õ¼ä, f,gÊÇVÉϵÄÏßÐԱ任(¼´f,g?L(V)),ÇÒfÓÐn¸ö»¥ÒìµÄÌØÕ÷¸ù.Ö¤Ã÷: fg?gfµÄ³äÒªÌõ¼þÊÇgÊÇ

f0?I (ºãµÈ±ä»»), f,f2,?,fn?1 µÄÏßÐÔ×éºÏ.

13.(ÖпÆÔº2010).ÉèMn(C)ÊǸ´ÊýÓòÉÏËùÓÐn½×·½Õ󹹳ɵÄÏßÐÔ¿Õ¼ä, T:Mn(C)?CÊÇÒ»¸öÏßÐÔÓ³Éä,Âú×ã

T(AB)?T(BA).

Ö¤Ã÷:´æÔÚ??C,ʹµÃ¶ÔÈÎÒâA?Mn(C),T(A)???Tr(A),ÆäÖÐ, Tr(A)±íʾ¾ØÕóAµÄ¼£.

32

14.(ÖпÆÔº2011).Éè?,?,?ÊÇ3άÏßÐÔ¿Õ¼äVµÄÒ»×é»ù,ÏßÐԱ任AÂú×ã

?A(??2???)??,??A(3??4?)??, ?A(4??5?)??.?ÊÔÇóAÔÚ»ù?,2???,?ϵľØÕó.

15.(Öпƴó1998).ÉèVÊÇʵÊýÓòÉÏÈ«Ìå2?2¾ØÕó×é³ÉµÄÏßÐÔ¿Õ¼ä(άÊýΪ4), VÉϵÄÏßÐԱ任?½«Ã¿¸ö¾ØÕóXÓ³µ½ËüµÄתÖÃX',ÇóVµÄÒ»×é»ù,ʹ?ÔÚÕâ×é»ùϵľØÕóΪ¶Ô½ÇÕó. 16.(Öпƴó2010).ÉèAΪÊýÓòFÉϵÄÏßÐÔ¿Õ¼äUµ½VÉϵÄÏßÐÔÓ³Éä.Ö¤Ã÷:

dimKer(A)?dimIm(A)?dimU.

17.(2009¾öÈü).ÉèVÊǸ´ÊýÓòCÉϵÄnάÏßÐÔ¿Õ¼ä,fj:V?CÊÇ·ÇÁãµÄÏßÐÔº¯Êý,j?1,2.Èô²»´æÔÚ0?c?CʹµÃf1?cf2,Ö¤Ã÷:ÈÎÒâµÄ??V¶¼¿É±íΪ???1??2ʹµÃ

f1(?)?f1(?2), f2(?)?f2(?1).

3.4 ÏßÐԱ任Óë¾ØÕóµÄÌØÕ÷ÎÊÌâ¡¢²»±ä×Ó¿Õ¼ä

1.(±±´ó1997).ÉèAÊÇʵÊýÓòRÉϵÄ3άÏßÐÔ¿Õ¼äVÄÚµÄÒ»¸öÏßÐԱ任,¶ÔVµÄÒ»×é»ù

?1,?2,?3,ÓÐ:

A?1?3?1?6?2?6?3, A?2?4?1?3?2?4?3, A?3??5?1?4?2?6?3.

(1)ÇóAµÄÈ«²¿ÌØÕ÷ÖµºÍÌØÕ÷ÏòÁ¿;(2)ÉèB?A?5A,ÇóBµÄÒ»¸ö·Çƽ·²µÄ²»±ä×Ó¿Õ¼ä. 2.(±±´ó2007).¸´¾ØÕóAÂú×㣺¶ÔÈÎÒâk,ÓÐTr(A)?0.ÊÔÇóAµÄÌØÕ÷Öµ.

3.(ÖпÆÔº2007).ÉèAÊǸ´ÊýÓòÉÏ6άÏßÐÔ¿Õ¼äVµÄÏßÐԱ任, AµÄÌØÕ÷¶àÏîʽΪ(??1)(??1)(??2),

Ö¤Ã÷VÄÜ·Ö½â³ÉÈý¸ö²»±ä×Ó¿Õ¼äµÄÖ±ºÍ,¶øÇÒËüÃǵÄάÊý·Ö±ðÊÇ1,2,3.

4.(ÖпÆÔº2003).¸ø¶¨RÉ϶þάÏßÐÔ¿Õ¼äVµÄÏßÐԱ任A,AÔÚÒ»×é»ùϵľØÕó±íʾΪ

32k31??0A???1?a0??,a?0.

??ÇóAµÄ²»±ä×Ó¿Õ¼ä.

5.(ÖпÆÔº2010).ÒÑÖª3½×Õý½»¾ØÕóAµÄÐÐÁÐʽΪ1.Ö¤Ã÷: AµÄÌØÕ÷¶àÏîʽһ¶¨Îª

f(?)??3?a?2?a??1,

33

ÆäÖÐ,aÊÇʵÊý,ÇÒ?1?a?3.

6.(ÖпÆÔº2010).Ö¤Ã÷:ÈÎÒân½×ʵ·½ÕóAµÄÌØÕ÷ÏòÁ¿Ò²ÊÇÆä°éËæ¾ØÕóA*µÄÌØÕ÷ÏòÁ¿.

7.(ÖпÆÔº2010).ÉèAÊÇnάʵÏßÐÔ¿Õ¼äVµÄÏßÐԱ任,Ö¤Ã÷: AÖÁÉÙÓÐÒ»¸öάÊýΪ1»ò2µÄ²»±ä×Ó¿Õ¼ä.

?c0??cn?1CC?8.(ÖпÆÔº2010).Éèn½×Ñ­»·¾ØÕóΪ????c?1Ó¦µÄÌØÕ÷ÏòÁ¿;(2)ÇóCµÄÐÐÁÐʽdet(C).

c1c0?c2?cn?1???cn?2?. (1)ÇóCµÄËùÓÐÌØÕ÷ÖµÒÔ¼°Ïà?????c0???ab?201129.(ÖпÆÔº2011).ÒÑÖª¶þ½×¾ØÕóA???cd??µÄÌØÕ÷¶àÏîʽΪ(??1),ÊÔÇóA?2011A.

??2?2??2??5?4?..(1)ÇóAµÄÌØÕ÷¶àÏîʽ,²¢È·¶¨ÆäÊÇ·ñÓÐÖØ10.(ÖпÆÔº2011).ÒÑÖª¾ØÕóA??2??2?45???¸ù;(2)ÇóÒ»¸öÕý½»¾ØÕóP,ʹµÃPAP?1Ϊ¶Ô½Ç¾ØÕó;(3)ÁîVÊÇËùÓÐÓëA¿É½»»»µÄʵ¾ØÕóÈ«Ìå,Ö¤Ã÷

VÊÇÒ»¸öʵÊýÓòÉϵÄÏßÐÔ¿Õ¼ä,²¢È·¶¨VµÄάÊý.

11.(ÖпÆÔº2011).ÉèA,BÊÇÁ½¸ön½×¸´¾ØÕón?1.(1)Èç¹ûAB?BA,Ö¤Ã÷A,BÓй«¹²µÄÌØÕ÷ÏòÁ¿;(2)Èç¹ûAB?BA??B,ÆäÖÐ?ÊÇÒ»¸ö·ÇÁ㸴Êý,ÄÇôA,BÊÇ·ñ»áÓй«¹²µÄÌØÕ÷ÏòÁ¿£¿»Ø´ð¡°ÊÇ¡±Çë¸ø³öÖ¤Ã÷;»Ø´ð¡°·ñ¡±Çë¸ø³ö·´Àý.

12.(Öпƴó1999).ÉèAÊǸ´ÊýÓòÉϵÄnάÏßÐÔ¿Õ¼äVÉϵÄÏßÐԱ任,¾ßÓÐn¸ö²»Í¬µÄÌØÕ÷Öµ

?1,?,?n,¶ø?1,?,?n·Ö±ðÊÇÊôÓÚÕâÐ©ÌØÕ÷ÖµµÄÌØÕ÷ÏòÁ¿.ÇóÖ¤: ?1????nÉú³ÉµÄÑ­»·×Ó¿Õ¼ä

µÈÓÚV.

13.(Öпƴó2010)ÉèÊýÓòFÉÏÓÐÏÞά¿Õ¼äVÉÏÏßÐԱ任AºÍBÂú×ãAB?aBA(a?F,a?1),ÇÒ

AÊÇ¿ÉÄæÏßÐԱ任,Ö¤Ã÷:(1)BΪÃÝÁã±ä»»(¼´´æÔÚÕýÕûÊýn,B?0).(2)AºÍBÓÐÒ»¸ö¹«¹²ÌØÕ÷

ÏòÁ¿.

14.(2009Ê¡ÇøÈü).¼ÙÉèVÊǸ´ÊýÓòCÉÏnάÏßÐÔ¿Õ¼ä(n?0),f,gÊÇVÉϵÄÏßÐԱ任,Èç¹û

nfg?gf?f,Ö¤Ã÷: fµÄÌØÕ÷Öµ¶¼ÊÇ0,ÇÒf,gÓй«¹²ÌØÕ÷ÏòÁ¿.

ËÄ¡¢Ë«ÏßÐÔº¯ÊýÓë¶þ´ÎÐÍ

34

4.1 Ë«ÏßÐÔº¯Êý

1.(±±´ó2000). (1)ÉèVÊÇʵÊýÓòÉϵÄÏßÐÔ¿Õ¼ä, fÊÇVÉϵÄÕý¶¨µÄ¶Ô³ÆË«ÏßÐÔº¯Êý, UÊÇVµÄÓÐÏÞά×Ó¿Õ¼ä.Ö¤Ã÷£ºV?U?U,ÆäÖÐU?????V:f(?,?)?0,???U?.(2) ÉèVÊÇÊýÓòKÉϵÄnά

?ÏßÐÔ¿Õ¼ä, gÊÇVÉϵķÇÍË»¯µÄ¶Ô³ÆË«ÏßÐÔº¯Êý. WÊÇVµÄ×Ó¿Õ¼ä.Áî

W?????V:g(?,?)?0,???W?,Ö¤Ã÷£º(a) dimV?dimW?dimW?;(bW?????W).

2.(±±´ó2007). fΪ˫ÏßÐÔº¯Êý,ÇÒ¶ÔÈÎÒâµÄ?,?,?¶¼ÓÐf(?,?)f(?,?)?f(?,?)f(?,?).ÊÔÖ¤Ã÷Ϊ¶Ô³ÆµÄ»ò·´¶Ô³ÆµÄ.

3.(2010Ê¡ÇøÈü).ÉèAΪn?nʵ¾ØÕó(δ±Ø¶Ô³Æ),¶ÔÈÎÒ»nάʵÏòÁ¿??(a1,?,an),?A?T?0 (Õâ

TÀï?±íʾ?µÄתÖÃ),ÇÒ´æÔÚnάʵÏòÁ¿?,ʹµÃ?A?T?0,ͬʱ¶ÔÈÎÒânάʵÏòÁ¿xºÍy,µ±Ê±

xAyT?0ÓÐxAyT?yAxT?0.Ö¤Ã÷:¶ÔÈÎÒânάʵÏòÁ¿v,¶¼ÓÐvA?T?0.

4.(2010¾öÈü).Éè?:Mn(R)?RÊÇ·ÇÁãÏßÐÔÓ³Éä,Âú×ã?(XY)??(YX),?X,Y?Mn(R),ÕâÀï

Mn(R)ÊÇʵÊýÓòRÉÏn½×·½Õó×é³ÉµÄÏßÐÔ¿Õ¼ä.ÔÚMn(R)É϶¨ÒåË«ÏßÐÔÐÍ(X,Y)??(XY).(1)Ö¤Ã÷

Ë«ÏßÐÔÐÍ(?,?)ÊÇ·ÇÍË»¯µÄ,¼´Èô(X,Y)?0,?Y?Mn(R),ÔòX?0;(2)Éè?A2n2k1?ÊÇMn(R)µÄÒ»×é»ù,

?B?

n2k1n?1,i?j,ÊÇÏàÓ¦µÄ¶Ôż»ù,¼´(Ai,Bj)??ij?? Ö¤Ã÷?AiBiÊÇÊýÁ¿¾ØÕó.

0,i?j.i?1?4.2 ¶þ´ÎÐÍ

1.(±±´ó2002).ÓÃÕý½»±ä»»»¯ÏÂÃæ¶þ´ÎÐÍΪ±ê×¼ÐΣº

22f(x1,x2,x3)?x12?x2?x3?4x1x2?4x1x3?4x2x3.

(ÒªÇóд³öÕý½»±ä»»µÄ¾ØÕóºÍÏàÓ¦µÄ±ê×¼ÐÎ).

2222.(ÖпÆÔº2007).Éè¶þ´ÎÐÍf?x1?x2?x3?2ax1x2?2x1x3?4bx2x3.ͨ¹ýÕý½»±ä»»»¯Îª±ê×¼22ÐÎf?y2,Çó²ÎÊýa,b¼°ËùÓõÄÕý½»±ä»». ?2y33.(ÖпÆÔº2006).ÒÑÖª¶þ´ÎÇúÃæ·½³Ìx?ay?z?2bxy?2xz?2yz?4¿ÉÒÔ¾­¹ýÕý½»±ä»»

222(x,y,z)T?P(x',y',z')T»¯ÎªÍÖÔ²ÖùÃæ·½³Ìy'2?4z'2?4..Çóa,bµÄÖµºÍÕý½»¾ØÕóP.

35

?a12??a2a14.(ÖпÆÔº2005).(1)¿¼ÂÇÈçÏÂÐÎʽµÄ¾ØÕóP?????aa?n1a1a22a2?ana2?a1an???a2an?,ÆäÖÐ,ai(1?i?n)¶¼????2??an?ΪʵÊý.Ö¤Ã÷:¾ØÕóP·Ç¸º¶¨;(2)Ö¤Ã÷:·ÇÁãʵ¶þ´ÎÐÍf(x1,?,xn)¿ÉÒÔд³É

f(x1,?,xn)?(u1x1???unxn)(v1x1???vnxn)

µÄ³äÒªÌõ¼þÊÇ£º»òÕßËüµÄÖÈΪ1,»òÕßËüµÄÖÈΪ2ÇÒ·ûºÅ²îΪ0.

2225.(Öпƴó2008).ÒÑÖª¶þ´ÎÐÍQ(x1,x2,x3)?x1?x2?x3?4x1x2?4x2x3?4x1x3.(1)ÓÃÕý½»±ä

»»½«Q(x1,x2,x3)»¯Îª±ê×¼ÐÎ.(2)ÅжÏÇúÃæQ(x1,x2,x3)?1µÄÀàÐÍ.

6.(Öпƴó2010).Ìî¿Õ:¶¨ÒåËùÓÐn½×ʵ·½Õ󹹳ɵÄʵÏßÐÔ¿Õ¼äVÉϵĶԳÆË«ÏßÐÔº¯ÊýΪ

f(X,Y)?Tr(XTY),X,Y?V,¶þ´ÎÐÍΪQ(X)?f(X,X),

ÔòQ(X)µÄÕý¡¢¸º¹ßÐÔÖ¸Êý·Ö±ðΪ .

4.3 ¶þ´ÎÐ͵ÄÓ¦ÓÃ.

1.(ÖпÆÔº2006).ÉèÓÐʵ¶þ´ÎÐÍf?xTAx, ÆäÖÐxÊÇxתÖÃ,AÊÇ3?3ʵ¶Ô³Æ¾ØÕó²¢Âú×ãÒÔÏ·½

32222³Ì£ºA?6A?11A?6I?0.ÊÔ¼ÆËã maxmaxf(x),ÆäÖÐ||x||2?x1?x2?x3,µÚÒ»¸ö¼«´óÖµÊÇ

A||x||?1TÂú×ãÒÔÉÏ·½³ÌµÄËùÓÐʵ¶Ô³Æ¾ØÕóAÀ´Çó.

222.(ÖпÆÔº2004).Áîf(x,y)?2x?7xy?y.,Çóf(x,y)ÔÚRÖе¥Î»Ô²Éϵļ«´óÖµÓ뼫Сֵ¼°¼«Öµµã.

22223.(Öпƴó1998).ÇóÖ¤:ÇúÃæ3x?2y.?2z?2xy?2xz?8?0ÊÇÍÖÇòÃæ.²¢Çó³öÕâ¸öÍÖÇòÃæËùΧ³É

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1.(±±´ó2008).(1) A,C·Ö±ðÊÇn½×ʵ¾ØÕó, BÊÇn?mʵ¾ØÕó,²¢ÇÒ???AT?BB??ÊÇÕý¶¨¾ØÕó.Ö¤Ã÷, ?C??Adet??BT?B???det(A)?det(C),²¢ÇҵȺųÉÁ¢µ±ÇÒ½öµ±B?0. C??2(2) A?(aij)n?nÊÇn½×ʵ¾ØÕó, |aij|?1,Ö¤Ã÷?det(A)??n2.

2.(ÖпÆÔº2007).ÉèAÊÇʵ¶Ô³Æ¾ØÕó,Èç¹ûAÊǰëÕý¶¨µÄ,Ôò´æÔÚʵµÄ°ëÕý¶¨¾ØÕóB,ʹµÃA?B.

36

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4.(ÖпÆÔº2004).ÉèA,BΪͬ½×¶Ô³ÆÕý¶¨Õó.ÈôA?B (¼´A?BΪÕý¶¨Õó),ÊÔÎÊÊÇ·ñÒ»¶¨ÓÐ

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7.(ÖпÆÔº2003).ÈôQΪn½×¶Ô³ÆÕý¶¨·½Õó,xΪnάʵÏòÁ¿,Ö¤Ã÷: 0?xT(Q?xxT)x?1,ÕâÀï

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8.(ÖпÆÔº2010).(1) n½×·½ÕóAÄܱí³É:A?H?K,ÆäÖÐH?H,K??K,BÊǾØÕóBµÄ¹²éîתÖÃ.Éèa,h,k´ú±íA,H,KÖÐÔªËØµÄ×î´óÄ£.Èôz?x?iy(x,y·Ö±ðÊÇʵ²¿ÓëÐ鲿)ÊÇAµÄÈÎÒ»ÌØÕ÷Öµ,ÊÔÖ¤Ã÷: |z|?na,|x|?nh,|y|?nk; (2)Ö¤Ã÷:Hermite¾ØÕóµÄÌØÕ÷Öµ¶¼ÊÇʵÊý;(3)Ö¤Ã÷:·´¶Ô³Æ¾ØÕóµÄÌØÕ÷Öµ¶¼ÊÇ´¿ÐéÊý.

9.(ÖпÆÔº2011).ÉèAÊÇn½×ʵ·½Õó,Ö¤Ã÷AΪʵ¶Ô³Æ¾ØÕóµ±ÇÒ½öµ±AA?A,ÆäÖÐA±íʾ¾ØÕóAµÄתÖÃ.

10.(Öпƴó2008). n½×ʵ¶Ô³Æ·½ÕóA,B,A?BµÄÕý¹ßÐÔÖ¸Êý·Ö±ðÊÇpA,pB,pA?B.Ö¤Ã÷:

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37

5.1 Euclid¿Õ¼äÓëSchmidtÕý½»»¯

1.(±±´ó1996).ÓÃR4[x]±íʾʵÊýÓòRÉÏ´ÎÊýСÓÚ4µÄÒ»Ôª¶àÏîʽ×é³ÉµÄ¼¯ºÏ,ËüÊÇÒ»¸öEuclid¿Õ¼ä,ÆäÉϵÄÄÚ»ý£º(f,g)?¸ö»ù.

?10f(x)g(x)dx.ÉèWÊÇÓÉÁã´Î¶àÏîʽ×é³ÉµÄ×Ó¿Õ¼ä, ÇóW?ÒÔ¼°ËüµÄÒ»

1??10???2?. (1)ÅжÏAÊÇ·ñΪÕý¶¨¾ØÕó,ÒªÇóд2.(±±´ó1999). ÉèʵÊýÓòÉϵľØÕóAΪ£ºA??06?1?22???³öÀíÓÉ;(2)ÉèVÊÇʵÊýÓòÉϵÄ3άÏßÐÔ¿Õ¼ä, VÉϵÄÒ»¸öË«ÏßÐÔº¯Êýf(?,?)ÔÚ

VµÄÒ»¸ö»ù?1,?2,?3ϵĶÈÁ¿¾ØÕóΪA. Ö¤Ã÷f(?,?)ÊÇVµÄÒ»¸öÄÚ»ý,²¢ÇÒÇó³öV¶ÔÓÚÕâ¸öÄÚ

»ýËù³ÉµÄÅ·ÊϿռäµÄÒ»¸ö±ê×¼Õý½»»ù.

3.(±±´ó2001).ÔÚʵÊýÓòÉϵÄnάÁÐÏòÁ¿¿Õ¼äRÖÐ,¶¨ÒåÄÚ»ýΪ(?,?)??'?,´Ó¶øR³ÉΪEuclid

nn?1?35?2????31?.ÇóÆë´ÎÏßÐÔ·½³Ì×éAX?0µÄ½â¿Õ¼äµÄÒ»¿Õ¼ä.(1)ÉèʵÊýÓòÉϵľØÕóA???21??1?79?4???¸öÕý½»»ù.(2)ÉèAÊÇʵÊýÓòRÉϵľØÕó,ÓÃW±íʾÆë´ÎÏßÐÔ·½³Ì×éAX?0µÄ½â¿Õ¼ä,ÓÃU±íʾA'µÄÁпռä(¼´, A'µÄÁÐÏòÁ¿×éÉú³ÉµÄ×Ó¿Õ¼ä).Ö¤Ã÷£ºU?W. 4.(±±´ó2005).£¨1£©ÉèʵÊýÓòRÉÏn¼¶¾ØÕóHµÄ(i,j)ԪΪ

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5.(ÖпÆÔº2005).¸ø¶¨Á½¸öËÄάÏòÁ¿?1?(1/3,?2/3,0,2/3)T,?2?(?2/6,0,1/6,1/6)T.Çó×÷Ò»¸öËĽ×Õý½»¾ØÕóQ,ÒÔ?1,?2×÷ΪËüµÄǰÁ½¸öÁÐÏòÁ¿.

6.(Öпƴó1997).ÉèRÊÇËùÓÐnάÐÐÏòÁ¿×é³ÉµÄnάŷÊϿռä,ÆäÖÐÏòÁ¿??(x1,?,xn)ºÍ

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38

n

µÄÕý½»²¹.

7.(Öпƴó1999).ÉèV?Rm?nÊÇʵÊýÓòÉÏÈ«Ìåm?nʵ¾ØÕó×é³ÉµÄÏòÁ¿¿Õ¼ä, SÊÇÕý¶¨µÄn½×ʵ

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9.(Öпƴó2010).ÉèVÊÇnάŷÊϿռä, (?,?)ΪÆäÄÚ»ý, V*ΪÆä¶Ôż¿Õ¼ä.Ö¤Ã÷£º(1)¶ÔÓÚÿ¸ö¸ø¶¨µÄ??V,Ó³Éäf?:V?R,??(?,?)ÊÇV*ÖеÄÒ»¸öÔªËØ.(2)Ó³Éäf:V?V*,??f?ÊÇ

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5.2 Å·ÊϿռäÖеÄÌØÊâÏßÐԱ任

1.(±±´ó1997).ÉèAÊÇάŷÊϿռäVÄÚµÄÒ»¸öÏßÐԱ任,Âú×㣺

(A?,?)??(?,A?) (??,??V).

(1)Èô?ÊÇAµÄÒ»¸öÌØÕ÷Öµ,Ö¤Ã÷ ??0;(2)Ö¤Ã÷VÄÚ´æÔÚÒ»×é±ê×¼Õý½»»ù,ʹAÔÚ´Ë×é»ùϵľØÕóΪ¶Ô½Ç¾ØÕó;(3)ÉèAÔÚVµÄij×é±ê×¼Õý½»»ùϵľØÕóΪA,Ö¤Ã÷£º°ÑA¿´×÷¸´ÊýÓòCÉϵÄn½×·½Õó,ÆäÌØÕ÷Öµ±ØÎª0»ò´¿ÐéÊý.

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39

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5.(±±´ó2010).ÏßÐԱ任AÊǶԳƱ任,ÇÒAÊÇÕý½»±ä»».Ö¤Ã÷AÊÇij¸ö¶ÔºÏ(¼´Âú×ãA?E,EÊǵ¥Î»±ä»»).

6.(±±´ó2010).VÊÇÄÚ»ý¿Õ¼ä,?,?ÊÇÖÐÁ½¸ö³¤¶ÈÏàµÈµÄÏòÁ¿,Ö¤Ã÷±Ø´æÔÚij¸öÕý½»±ä»»,½«?±äµ½?. 7.(ÖпÆÔº2003).ÉèAÊÇÅ·ÊϿռäRµÄÒ»¸ö±ä»».ÊÔÖ¤:Èç¹ûA±£³ÖÄÚ»ý²»±ä,¼´¶ÔÓÚRÖÐÈÎÒâÁ½¸öÏòÁ¿?,?¶¼ÓÐ(A?,A?)?(?,?), ÄÇô,ËüÒ»¶¨ÊÇÏßÐÔµÄ,¶øÇÒÊÇÕý½»µÄ.

8.(Öпƴó1997).ÉèAÊÇnάŷÊϿռäVµÄÏßÐԱ任,³ÆAÊǶԳƵÄ,Èç¹û¶ÔÈÎÒâ

nn2?,??V,(A?,?)?(?,A?).³Æ¶Ô³ÆÏßÐԱ任AÊǷǸºµÄ,Èç¹û¶ÔÈÎÒâ??V,(?,Aa)?0.Ö¤Ã÷:

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40

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1.(±±´ó1996). n¼¶¾ØÕóA³ÆÎªÖÜÆÚ¾ØÕó,Èç¹û´æÔÚÕýÕûÊýmʹA?I,ÆäÖÐIÊǵ¥Î»¾ØÕó,Ö¤Ã÷:¸´ÊýÓòÉϵÄÖÜÆÚ¾ØÕóÒ»¶¨¿ÉÒԶԽǻ¯.

3.(±±´ó1998).ÓÃJ±íÊ¾ÔªËØÈ«Îª1µÄn¼¶¾ØÕó,n?2.Éèf(x)?a?bxÊÇÓÐÀíÊýÓòÉϵÄÒ»Ôª¶àÏîʽ,ÁîA?f(J).(1)ÇóJµÄÈ«²¿ÌØÕ÷ÖµºÍÈ«²¿ÌØÕ÷ÏòÁ¿;(2)ÇóAµÄËùÓÐÌØÕ÷×Ó¿Õ¼ä;(3)AÊÇ·ñ¿ÉÒԶԽǻ¯£¿Èç¹û¿É¶Ô½Ç»¯,Çó³öÓÐÀíÊýÓòÉϵÄÒ»¸ö¿ÉÄæ¾ØÕóPʹµÃPAPΪ¶Ô½Ç¾ØÕó,²¢ÇÒд³öÕâ¸ö¶Ô½ÇÕó.

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41

?100???nn?2?A2?I,²¢¼ÆËãA100,ÆäÖÐ7.(ÖпÆÔº2007).ÒÑÖªA??101?,ÊÔÖ¤Ã÷¶Ôn?3,ÓÐA?A?010???I±íʾµ¥Î»¾ØÕó.

8.(ÖпÆÔº2005).Ö¤Ã÷£º(1)ÈκÎn½×ʵ¶Ô³Æ·½ÕóA±ØºÏͬÓÚ¶Ô½ÇÕóD?diag{?1,?,?n},¼´´æÔÚn½×·ÇÆæÒìʵ·½ÕóCʹµÃCAC?D,ÕâÀï?i??1»ò0»ò1;(2)ÈκÎn½×ʵ·´¶Ô³Æ·ÇÆæÒì·½ÕóB±ØÎªÅ¼Êý½×(¼´n?2k),ÇÒºÏͬÓÚ¿é¶Ô½ÇÕóF?diag{J1,?,Jm},¼´´æÔÚn½×·ÇÆæÒìʵ·½ÕóEʹµÃ

T?0?1?ETBE?F,ÕâÀïJk???10??;(3)¶Ô¼£(¶Ô½ÇÔªÖ®ºÍ)Ϊ0µÄn½×ʵ·½ÕóG,±Ø´æÔÚʵÕý½»ÕóH,

??ʹµÃHGHµÄÖ÷¶Ô½ÇԪȫΪÁã.(×¢£ºÕâÀïX±íʾXµÄתÖÃ).

7.(ÖпÆÔº2003).ÉèAÊÇ2003½×ʵ·½Õó,ÇÒA?0,ÕâÀïrÊÇ×ÔÈ»Êý.ÎÊAµÄÖÈrank(A)×î´óÊǶàÉÙ£¿

8.(Öпƴó1998).ÉèA,BÊÇn½×¸´·½Õó,ÇÒAB?BA.Ö¤Ã÷:(1)A,BÓй«¹²µÄÌØÕ÷ÏòÁ¿;(2)Èç¹û

rTTA,B¶¼ÏàËÆÓÚ¶Ô½ÇÕó,Ôò´æÔÚͬһ¸ö¿ÉÄæ¸´·½ÕóT,ʹT?1ATÓëT?1BTͬʱΪ¶Ô½ÇÕó.

8.(Öпƴó2008).Ö¤Ã÷: n½×ʵ·½ÕóAÕý½»ÏàËÆÓÚÒ»¸ö×¼ÉÏÈý½ÇÕó

?B1???B??0?0????0ÆäÖÐBjΪ¶þ½×ʵ·½Õó,?kΪʵÊý.

?**?*?????0???Bs?0???0?2s?1?0*????*?, *?????n?9.(Öпƴó2008).Éèʵ·½ÕóA,BÏàËÆÇÒÏàºÏ,ÎÊA,BÊÇ·ñÕý½»ÏàËÆ,ÊÔÖ¤Ö®.

10.(2010¾öÈü).ÉèA?Mn(C),¶¨ÒåÏßÐԱ任?A:Mn(C)?Mn(C),?A(X)?AX?XA.Ö¤Ã÷:µ±A¿É¶Ô½Ç»¯Ê±, ?AÒ²¿É¶Ô½Ç»¯,ÕâÀïMn(C)ÊǸ´ÊýÓòCÉÏn½×·½Õó×é³ÉµÄÏßÐÔ¿Õ¼ä.

6.2 Jordan±ê×¼Ðμ°ÆäÓ¦ÓÃ

1.(ÖпÆÔº2006). A?R2006?2006ÊǸø¶¨µÄÃÝÁãÕó(¼´£º´æÔÚÕýÕûÊýp,ʹµÃA?0¶øApp?1?0),ÊÔ

·ÖÎöÏßÐÔ·½³ÌAx?0(x?R2006)·ÇÁã¶ÀÁ¢½â¸öÊýµÄ×î´óÖµÓë×îСֵ.

42

?0??02.(ÖпÆÔº2005).(1)Çó¾ØÕóA??0??0?100011001??1?AeµÄJordan±ê×¼ÐÎ,²¢¼ÆËã (×¢£º°´Í¨³£¶¨Òå?1?0??4.5?1??4??1213A2005B??3?3.51e?I?A?A?A??);(2)Éè??£¬ ÇóB (¾«È·µ½Ð¡Êýµãºó4λ).

2!3!??2?31.5???3.(ÖпÆÔº2011).ÉèAÊÇn½×ʵ·½Õó,ÆäÌØÕ÷¶àÏîʽÓÐÈçÏ·ֽ⣺p(?)?det(?E?A)??(???k?1sk)rk,

ÆäÖÐEΪn½×µ¥Î»·½Õó,Öî?kÁ½Á½²»ÏàµÈ.ÊÔÖ¤Ã÷AµÄJordan±ê×¼ÐÎÖÐÒÔ?kÎªÌØÕ÷ÖµµÄJordan¿éµÄ¸öÊýµÈÓÚÌØÕ÷×Ó¿Õ¼äV?kµÄάÊý.

4.(Öпƴó1999).ÉèFn[x]ÊÇÊýÓòFÉÏ´ÎÊý?nµÄÈ«Ìå¶àÏîʽ¹¹³ÉµÄÏßÐÔ¿Õ¼ä. Fn[x]ÉÏÏßÐԱ任

D½«Ã¿¸ö¶àÏîʽf(x)Ó³µ½Æäµ¼º¯Êýf'(x).(1)ÇóDµÄÌØÕ÷¶àÏîʽºÍ×îС¶àÏîʽ.(2)ÕÒ³öFn[x]µÄ

Ò»×é»ù,ʹDÔÚÕâ×é»ùϵľØÕóÊÇJordan±ê×¼ÐÎ.(3)ÉèIÊÇFn[x]Éϵĵ¥Î»±ä»»,

DkA?I??.ÇóÖ¤AÊÇFn[x]ÉϵĿÉÄæ±ä»»,²¢Çó³öAµÄÄæ.

k?1k!n?15.(Öпƴó2008).Ìî¿Õ£º(1)ÒÑÖªËĽ×?·½ÕóA(?)µÄÖÈΪ4,³õµÈÒò×Ó×éΪ

?,?2,?3,(??1)2,(??1)3,??1,

ÔòA(?)µÄ²»±äÒò×ÓÊÇ ,ÐÐÁÐʽÒò×ÓÊÇ .

?00?2??????0?,ÇóËüµÄSmith±ê×¼ÐÎ . (2) A(?)??0??2??0?0???1??0(3) A??0??0?0?0100??1010?0101?µÄJordan±ê×¼ÐÎÊÇ . ?0010?0001?? 43

???1??2?1???226.(Öпƴó2010).Ìî¿Õ: ??¾ØÕó?3??1??2?3??1?µÄSmith±ê×¼ÐÎÊÇ .

???1?22???1???2?11????17.(Öпƴó2010).ÉèA??22?1?,Çó·½ÕóP,ʹµÃPAPΪAµÄJordan±ê×¼ÐÎ.

?12?1????01030???28.(2010Ê¡ÇøÈü).ÉèB??002010?.Ö¤Ã÷X?BÎÞ½â,ÕâÀïXΪÈý½×δ֪¸´·½Õó.

?000???

6.3 ×îС¶àÏîʽ

?110???1.(±±´ó1999).ÉèʵÊýÓòÉϵľØÕóAΪ£ºA???101?. (1)ÇóAµÄÌØÕ÷¶àÏîʽfA(?);(2)

??300???fA(?)ÊÇ·ñΪʵÊýÓòÉϵIJ»¿ÉÔ¼¶àÏîʽ;(3)ÇóAµÄ×îС¶àÏîʽ,ÒªÇóд³öÀíÓÉ;(4)ʵÊýÓòÉϵľØÕó

AÊÇ·ñ¿É¶Ô½Ç»¯,ÒªÇóд³öÀíÓÉ.

2.(±±´ó2006).(1)ÉèAÊÇʵÊýÓòRÉϵÄn½×¶Ô³Æ¾ØÕó,ËüµÄÌØÕ÷¶àÏîʽf(?)µÄËùÓв»Í¬¸´¸ùΪʵÊý: ?1,?2,?,?n. °ÑAµÄ×îС¶àÏîʽm(?)·Ö½â³ÉÉϲ»¿ÉÔ¼¶àÏîʽµÄ³Ë»ý,˵Ã÷ÀíÓÉ;(2) ÉèAÊÇʵÊýÓòRÉϵÄn½×¶Ô³Æ¾ØÕó,ÁîA(?)?A?,???R. ¸ù¾ÝµÚ(1)ÎÊÖÐm(?)µÄÒòʽ·Ö½â,°Ñ·Ö½â

nRn³ÉÏßÐԱ任AµÄ²»±ä×Ó¿Õ¼äµÄÖ±ºÍ, ˵Ã÷ÀíÓÉ.

3.(Öпƴó1998).Ö¤Ã÷:¸´·½ÕóAµÄ×îС¶àÏîʽÓëÌØÕ÷¶àÏîʽÏàµÈµÄ³äÒªÌõ¼þÊÇ:AµÄÌØÕ÷×Ó¿Õ¼ä

¶¼ÊÇһάµÄ.

Æß¡¢¶àÏîʽ»·

7.1 Ò»Ôª¶àÏîʽµÄÕû³ýÀíÂÛ

1.(±±´ó2000).Éèf(x)ºÍp(x)¶¼ÊÇÊ×ÏîϵÊýΪµÄÕûϵÊý¶àÏîʽ,ÇÒp(x)ÔÚÓÐÀíÊýÓòQÉϲ»¿ÉÔ¼.Èç¹ûp(x)Óëf(x)Óй«¹²¸´¸ù?,Ö¤Ã÷(1)ÔÚQ[x]ÖÐ,p(x)Õû³ýf(x);(2)´æÔÚÊ×ÏîϵÊýΪ1µÄÕûϵÊý¶àÏîʽg(x),ʹµÃf(x)?p(x)g(x). 2.(±±´ó2002).¶ÔÓÚÈÎÒâ·Ç¸ºÕûÊýn,Áîfn(x)?xn?2?(x?1)2n?1, Ö¤Ã÷: x2?x?1,fn(x)?1.

?? 44

3.(ÖпÆÔº2007).Éè¶àÏîʽf(x),g(x),h(x)Ö»ÓзÇÁã³£Êý¹«Òò×Ó,Ö¤Ã÷:´æÔÚ¶àÏîʽ

u(x),v(x),w(x),ʹµÃu(x)f(x)?v(x)g(x)?w(x)h(x)?1.

4.(ÖпÆÔº2007).Éèm,n,p¶¼ÊǷǸºÕûÊý,Ö¤Ã÷£ºx2?x?1|x3m?x3n?1?x3p?2.

5.(ÖпÆÔº2005).ÊÔÇó7´Î¶àÏîʽf(x),ʹf(x)?1Äܱ»(x?1)4Õû³ý,¶øf(x)?1Äܱ»(x?1)4Õû³ý.

7.2 ¸ùÓë¿ÉÔ¼ÐÔ

1.(±±´ó1997).Éèf(x)ÊÇÓÐÀíÊýÓòQÉϵÄÒ»¸öm´Î¶àÏîʽ(m?0),nÊÇ´óÓÚmµÄÕýÕûÊý,Ö¤Ã÷

n2²»ÊÇf(x)µÄʵ¸ù.

2.(±±´ó2000).Éèf(x)ºÍp(x)¶¼ÊÇÊ×ÏîϵÊýΪµÄÕûϵÊý¶àÏîʽ,ÇÒp(x)ÔÚÓÐÀíÊýÓòQÉϲ»¿ÉÔ¼.Èç¹ûp(x)Óëf(x)Óй«¹²¸´¸ù?,Ö¤Ã÷(1)ÔÚQ[x]ÖÐ,p(x)Õû³ýf(x);(2)´æÔÚÊ×ÏîϵÊýΪ1µÄÕûϵÊý¶àÏîʽg(x),ʹµÃf(x)?p(x)g(x)..

3.(±±´ó2008).f(x)ΪһÕûϵÊý¶àÏîʽ,n²»ÄÜÕû³ýf(0),f(1),?,f(n?1),Ö¤Ã÷f(x)ÎÞÕûÊý¸ù.

4.(±±´ó2010).ÕûϵÊý¶àÏîʽf(x)??ak?0nkxk(n?2010),Èô´æÔÚËØÊýpÂú×ã:

i) an²»±»pÕû³ý, ¢¢) p|ak,k?0,1,2,?,2008; ¢£) p2|a0.

Ö¤Ã÷f(x)±ØÓдÎÊý²»µÍÓÚ2009µÄ²»¿ÉÔ¼ÕûϵÊýÒòʽ.

npk5.(ÖпÆÔº2011).ÉèÊǼÈÔ¼·ÖÊý, f(x)??akxÊÇÕûϵÊý¶àÏîʽ,¶øÇÒ

qk?0?p?(1) p|a0,f??q???0.Ö¤Ã÷£º

??¶øq|an;(2)¶ÔÈÎÒâÕûÊým,ÓÐ(p?mq)|f(m).

7.3 ʵϵÊý¶àÏîʽʵ¸ùµÄ·Ö²¼ 7.4 ¶àÔª¶àÏîʽ

°Ë¡¢½âÎö¼¸ºÎ

8.1 ÏòÁ¿Óë×ø±ê

1.(±±´ó2001). ÔÚ¿Õ¼äÖ±½Ç×ø±êϵÖÐ,µãA,B,CµÄ×ø±êÒÀ´ÎΪ£º(?2,1,4),(?2,?3,?4),(?1,3,3) (1)ÇóËÄ

45

ÃæÌåOABCµÄÌå»ý;(2)ÇóÈý½ÇÐÎABCµÄÃæ»ý.

2.(±±´ó2006).Ö¤Ã÷ËÄÃæÌåµÄÿ¸ö¶¥µãµ½¶ÔÃæÖØÐĵÄÁ¬Ïß¶¼ÏཻÓÚÒ»µã,¶øÇҸõã·ÖÏ߶αÈΪ3:1.

x2y2z2???1±»µã(2,?1,1)ƽ·ÖµÄÏÒ. 3.(±±´ó2007).ÇóÍÖÇòÃæ

251694.(Öпƴó1998,2008). ÇóÒÔA(1,2,2),B(2,4,1),C(1,?3,5),D(4,?2,3)Ϊ¶¥µãµÄËÄÃæÌåµÄÌå»ý. 5.(Öпƴó1999).ÉèÆ½Ãæ?:Ax?By?Cz?D?0ÓëÁ¬½ÓÁ½µãM1(x1,y1,z1)ºÍM2(x2,y2,z2)µÄÏß¶ÎÏཻÓÚµãM,ÇÒM1M?kMM2.Ö¤Ã÷ k?Ax1?By1?Cz1?D.

Ax2?By2?Cz2?D?????????????6.(2009¾öÈü).Ìî¿Õ: Éè?a?b??c?6,Ôò?a?b??(b?c)?(a?c)? .

????

8.2 ¹ì¼£Óë·½³Ì

1.(±±´ó1997).¹ýxÖáºÍyÖá·Ö±ð×ö¶¯Æ½Ãæ,½»½Ç?Êdz£Êý,Çó½»Ï߹켣µÄ·½³Ì,²¢ÇÒÖ¤Ã÷ËüÊÇÒ»¸ö×¶Ãæ.

2.(±±´ó1998). ÔÚÖ±½Ç×ø±êϵÖÐ,ÇòÃæµÄ·½³ÌΪ£º(x?1)2?y2?(z?1)2?4.ÇóËùÓÐÓëÏòÁ¿

u(1,1,1)ƽÐеÄÇòÃæµÄÇÐÏßËù¹¹³ÉµÄÇúÃæµÄ·½³Ì.

3.(±±´ó2001). ÔÚ¿Õ¼äÖ±½Ç×ø±êϵÖÐ, l1:x?ayzxy?1z??Óël2:??,ÊÇÒ»¶ÔÏֱཻ1?2321?2Ïß(1)Çóa;(2)Çól2ÈÆl1Ðýת³öµÄÇúÃæµÄ·½³Ì.

?z?3x?2,4.(±±´ó2002). ÔÚ¿Õ¼äÖ±½Ç×ø±êϵÖÐ,ÇóÖ±Ïß? ÈÆzÖáÐýתËùµÃÐýתÇúÃæµÄ·½³Ì.

z?2y?1?5.(±±´ó2010). ÇóÖ±Ïß??x?y?z?1, ÈÆzÖáÐýתËùµÃÐýתÇúÃæµÄ·½³Ì,²¢Ö¸³öÕâÊÇʲôÇúÃæ.

?x?y?z?1?2x?y?z?06. (±±´ó2005).ÔÚÖ±½Ç×ø±êϵÖУ¬ÇóÖ±Ïßl:?µ½Æ½Ãæ?:3x?By?z?0µÄÕý½»Í¶

x?y?2z?1?Ó°¹ì¼£µÄ·½³Ì¡£ÆäÖÐBÊdz£Êý.

7.(±±´ó2006).Ò»ÌõÖ±ÏßÓë×ø±êÆ½ÃæyOzÃæ¡¢zOxÃæ¡¢xOyÃæµÄ½»µã·Ö±ðÊÇA,B,C.µ±Ö±Ï߱䶯ʱ,Ö±ÏßÉϵÄÈý¸ö¶¨µãA,B,CÒ²·Ö±ðÔÚ×ø±êÆ½ÃæÉϱ䶯.´ËÍâ,Ö±ÏßÉÏÓеÚËĵãP,µãPµ½ÈýµãµÄ¾àÀë·Ö±ðÊÇa,b,c.Çó¸ÃÖ±Ïß°´ÕÕ±£³ÖµãA,B,C·Ö±ðÔÚ×ø±êÆ½ÃæÉϵĹæÔòÒÆ¶¯Ê±,µãPµÄ¹ì¼£.

?8.(±±´ó2006). PÊÇÇòÄÚÒ»¶¨µã, A,B,CÊÇÇòÃæÉÏÈý¶¯µã, ?APB??BPC??CPA?2,ÒÔ

46

PA,PB,PCΪÀâ×÷ƽÐÐÁùÃæÌå,¼ÇÓëPÏà¶ÔµÄ¶¥µãΪQ,ÇóQµãµÄ¹ì¼£.

9.(±±´ó2008).Ö¤Ã÷ÇòÃæx2?y2?z2?2x?2y?4z?2?0ÓëÇòÃæx2?y2?z2?2x?6y?1?0Óн»µã,²¢Çó³ö½»Ô²µÄÔ²ÐÄ×ø±ê.

10.(Öпƴó2008).ÒÑÖªÖ±Ïßl1:x?1?y?2??z?1ºÍÆ½Ãæ?:2x?y?z?0. ?2(1)Çól1ÔÚ?ÉϵÄͶӰֱÏßl2µÄ·½³Ì.(2)Çól1ÈÆl2ÐýתËùµÃµÄÐýתÇúÃæµÄ·½³Ì.

?y?x211.(Öпƴó2010).Ìî¿Õ£º(1)ÒÔÇúÏß?Ϊ׼Ïß,Ô­µãΪ¶¥µãµÄ×¶Ãæ·½³ÌΪ . (2)

?z?2ÒÔxOyÆ½ÃæÉϵÄÇúÏßf(x,y)?0ÈÆxÖáÐýתËùµÃµÄÐý×ªÃæµÄ·½³ÌÊÇ £» Èç¹ûÇúÏß·½³ÌÊÇx2?y2?1?0,Óɴ˵õ½µÄÇúÃæÀàÐÍÊÇ . xyz?b??. (1)ÎÊ:²ÎÊýa,bÂú×ãʲô1a1Ìõ¼þʱ, LÓëL'ÊÇÒìÃæÖ±Ïߣ¿(2)µ±LÓëL'²»ÖغÏʱ,ÇóL'ÈÆLÐýתËùÉú³ÉµÄÐý×ªÃæ?µÄ·½³Ì,

12.(2009¾öÈü).ÒÑÖªÁ½Ö±Ïߵķ½³Ì:L:x?y?z, L':²¢Ö¸³öÇúÃæ?µÄÀàÐÍ.

8.3 Æ½ÃæÓë¿Õ¼äÖ±Ïß

1.(±±´ó1996).ÔÚ·ÂÉä×ø±êϵÖУ¬Çó¹ýµãM0(0,0,?2),ÓëÆ½Ãæ?1:3x?y?2z?1?0ƽÐÐ,ÇÒÓëÖ±Ïßl1:x?1y?3z??ÏཻµÄÖ±ÏßlµÄ·½³Ì. 4?212.(±±´ó1998,2002).ÉèÔÚÖ±½Ç×ø±êϵÖиø³öÁËÁ½Ìõ»¥ÏàÒìÃæµÄÖ±Ïßl1,l2µÄÆÕͨ·½³Ì£º

?x?y?z?1?0, l2l1:??x?y?2z?1?0;?3x?y?1?0, :??y?3z?2?0.(1)¹ýl1×÷Æ½Ãæ?,ʹ?Óël1ƽÐÐ;(2)Çól1Óël2¼äµÄ¾àÀë;(3)Çól1Óël2µÄ¹«´¹Ïߵķ½³Ì. 3.(±±´ó1999).ÔÚ·ÂÉä×ø±êϵÖУ¬ÒÑÖªÖ±Ïßl1,l2µÄ·½³Ì·Ö±ðÊÇ

l1:x?13y?5zx?10y?7z??, l2:??, 231541(1)ÅжÏÖ±Ïßl1Óël2µÄλÖùØÏµ,ÒªÇóд³öÀíÓÉ;(2)ÉèÖ±ÏßlµÄÒ»¸ö·½ÏòÏòÁ¿Îªv(8,7,1),²¢ÇÒlÓël1ºÍl2¶¼Ïཻ,ÇóÖ±ÏßlµÄ·½³Ì.

x2?y2?z2?1µÄ½»ÏßΪһ¸öÔ². 4.(±±´ó2006).ÇóÒ»¸ö¹ýxÖáµÄÆ½Ãæ,ʹµÃÆäÓëµ¥Ò¶Ë«ÇúÃæ4 47

5.(±±´ó2006).ÔÚ·ÂÉä×ø±êϵÖУ¬ÒÑÖªÖ±Ïßl1µÄ·½³ÌΪ???x?y?z?7?0, l2¹ýµãM(?1,1,2),

?2x?y?6?0,ƽÐÐÓÚÏòÁ¿u(1,2,?3),ÅжÏÖ±Ïßl1Óël2µÄλÖùØÏµ,ÒªÇóд³öÀíÓÉ.

?A1x?B1y?C1z?D1?0,6.(±±´ó2007). lÖ±Ïߵķ½³ÌΪ?ÎÊϵÊýÒªÂú×ãʲôÌõ¼þ,²ÅÄÜʹµÃÖ±ÏßÂú

Ax?By?Cz?D?0.222?2×ãÏÂÁÐÌõ¼þ(1)¹ýÔ­µã;(2)ƽÐÐÓÚxÖá,µ«²»ÓëxÖáÖØºÏ;(3)ÓëyÖáÏཻ;(4)ÓëzÖáÖØºÏ. 7.(±±´ó2008).Çó¹ýÖ±Ïßl:??x?y?z?4?0,ÇÒÓëÆ½Ãæ?1:x?y?2z?0´¹Ö±µÄÆ½Ãæ?2.

?x?y?3z?0,??x?y?z?0,8.(±±´ó2008).Ö±Ïßl1¹ýµã(1,1,1),,²¢ÇÒÓëÖ±Ïßl2:?Ïཻ,½»½ÇΪ,Çól1µÄ·½³Ì.

3?x?y?3z?0,9.(±±´ó2010).Çó¹ýzÖáÇÒÓëÆ½Ãæx?2y?3z?1¼Ð½Ç10.(Öпƴó2010).Éè¿Õ¼äÉÏÓÐÖ±Ïßl1:?3,ΪµÄÆ½ÃæµÄ·½³Ì.

x?1yz??ºÍl2:(x,y,z)?(3?2t,t,3t?3).ÉèÆ½Ãæ?310ÓëÖ±Ïßl1,l2ƽÐÐ,ÇÒ?Óël1µÄ¾àÀëÊÇ91,Çó?µÄ·½³Ì.

8.4 ¶þ´ÎÇúÃæ

1.(±±´ó1996).×÷Ö±½Ç×ø±ê±ä»»,°ÑÏÂÊö¶þ´ÎÇúÃæ·½³Ì»¯³É±ê×¼·½³Ì,²¢Ö¸³öËüÊÇʲôÇúÃæ£º

x2?4y2?z2?4xy?8xz?4yz?2x?y?2z?25?0 162.(±±´ó1999). ÔÚÖ±½Ç×ø±êϵOxyzÖÐ,Éè¶¥µãÔÚÔ­µãµÄ¶þ´Î×¶ÃæSµÄ·½³ÌΪ

a11x2?a22y2?a33z2?2a12xy?2a13xz?2a23yz?0.

(1)Èç¹ûÈýÌõ×ø±êÖá¶¼ÊÇSµÄĸÏß,Çóa11,a22,a33;(2)Ö¤Ã÷£ºÈç¹ûSÓÐÈýÌõ»¥Ïà´¹Ö±µÄֱĸÏß,Ôò

a11?a22?a33?0.

3.(±±´ó2000). ÔÚÖ±½Ç×ø±êϵÖÐ,Ò»¸öÖùÃæµÄ×¼Ïß·½³ÌΪ??xy?4,ĸÏß·½ÏòΪ(1,?1,1)ÇóÕâ¸öÖùÃæµÄ·½³Ì.

?z?0,?204????14.(±±´ó2000).(1)ÉèʵÊýÓòÉϵľØÕóA??060?, ÇóÕý½»¾ØÕóT, ʹµÃTATΪ¶Ô½Ç¾ØÕó,

?402???²¢ÇÒд³öÕâ¸ö¶Ô½ÇÕó.(2) ÔÚÖ±½Ç×ø±êϵOxyzÖÐ,¶þ´ÎÇúÃæSµÄ·½³ÌΪ

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x2y25.(±±´ó2007).Ö¤Ã÷Ë«ÇúÅ×ÎïÃæ2?2?2zµÄÏ໥´¹Ö±µÄֱĸÏߵĽ»µãÔÚË«ÇúÏßÉÏ.

ab6.(±±´ó2008).Æ½ÃæAx?By?Cz?D?0Óëµ¥Ò¶Ë«ÇúÃæx2?y2?z2?1µÄ½»ÏßÊÇÁ½ÌõÖ±Ïß,Ö¤Ã÷£ºA?B?C?D.

7.(2009Ê¡ÇøÈü).Çó¾­¹ýÈýƽÐÐÖ±ÏßL1:x?y?z,L2:x?1?y?z?1,L3:x?y?1?z?1, µÄÔ²ÖùÃæ·½³Ì.

8.(2010Ê¡ÇøÈü).ÒÑÖª¶þ´ÎÇúÃæ?(·ÇÍË»¯)¹ýÒÔϾŵ㣺A(1,0,0),B(1,1,2),C(1,?1,?2),D(3,0,0),

2222E(3,1,2),F(3,?2,?4),G(0,1,4),H(3,?1,?2),I(5,22,8).ÎÊ?ÊÇÄÄÒ»ÀàÇúÃæ£¿

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8.5 ¶þ´ÎÇúÏßµÄÒ»°ãÀíÂÛ

1.(±±´ó1997).ÅжÏÏÂÁжþ´ÎÇúÏßÀàÐÍ£º(1) x2?3xy?y2?10x?10y?21?0; (2)x?4xy?4y?20x?10y?30?0.

2.(±±´ó2000). ÔÚÆ½ÃæÖ±½Ç×ø±êϵOxyÖÐ,¶þ´ÎÇúÏߵķ½³ÌΪ£º

22x2?3xy?y2?10x?10y?21?0.

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3.(±±´ó2005).ÔÚÖ±½Ç×ø±êϵÖжÔÓÚ²ÎÊý?µÄ²»Í¬È¡Öµ£¬ÅжÏÏÂÃæÆ½Ãæ¶þ´ÎÇúÏßµÄÐÎ×´£º

x2?y2?2?xy???0.¶ÔÓÚÖÐÐÄÐÍÇúÏߣ¬Ð´³ö¶Ô³ÆÖÐÐĵÄ×ø±ê£»¶ÔÓÚÏßÐÄÐÍÇúÏߣ¬Ð´³ö¶Ô³ÆÖ±

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4.(±±´ó2010).¶¨Òå·ÂÉä×ø±êϵOxyÖеÄÒ»¸ö±ä»»f:?(1)ÇóÔÚfϵIJ»±äÖ±Ïß.

(2)ÒÔÁ½Ìõ²»±äÖ±ÏßÎª×ø±êÖὨÁ¢·ÂÉä×ø±êϵO'x'y',ÇóÔÚ´Ë×ø±êϵÖÐfµÄ±ä»»¹«Ê½.

(3)Óò»¹ýÔ²×¶¶¥µãµÄÆ½ÃæÇиîÔ²×¶,Ö¤Ã÷Ëù½ØµÄÇúÏßÖ»¿ÉÄÜΪÍÖÔ²¡¢Ë«ÇúÏߺÍÅ×ÎïÏß.²¢ËµÃ÷ÇúÏß

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5.(Öпƴó1997).¸ø¶¨Ö±½Ç×ø±êÆ½ÃæOxyÉϵĶþ´ÎÇúÏß?,Æä·½³ÌΪ

Ax2?Bxy?Cy2?Dx?Ey?F?0.

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B2?4ACB/2D/2??A??24AC?BCE/2?. ?0µÄ¸ù,¶ø??det?B/2ÆäÖÐ?1ºÍ?2ÊÇ·½³Ì?1?(A?C)??4?D/2E/2F????1??x??26.(Öпƴó1999).ÔÚÆ½ÃæÖ±½Ç×ø±êϵÖÐ, ??y????3?????23??x?2????±íÊ¾Æ½ÃæÉϵĵãµÄÔõÑùµÄ±ä»»£¿ ???1?y??2?7.(Öпƴó2010).Ìî¿Õ£º¶þ´ÎÇúÏßx2?4xy?y2?10x?10y?21?0µÄÀàÐÍÊÇ£º ͨ¹ýתÖáÈ¥µô½»²æÏîµÄת½Ç½Ç¶ÈÊÇ £¨Ö»ÐèÒªÌîдһ¸ö½Ç¶È¼´¿É£©.

50